# Symmetric Pseudo-Random Matrices

## TL;DR

This paper presents a simple explicit method to generate symmetric pseudo-random sign matrices whose spectra approximate Wigner's semicircular law, using Golomb sequences, with low Kolmogorov complexity.

## Contribution

It introduces a new explicit construction of symmetric pseudo-random matrices based on Golomb sequences, with spectra converging to the semicircular law and minimal Kolmogorov complexity.

## Key findings

- Spectra of constructed matrices converge to Wigner's semicircular law
- Construction is explicit and based on Golomb sequences
- Kolmogorov complexity is at most 2log(n) bits

## Abstract

We consider the problem of generating symmetric pseudo-random sign (+/-1) matrices based on the similarity of their spectra to Wigner's semicircular law. Using binary m-sequences (Golomb sequences) of lengths n=2^m-1, we give a simple explicit construction of circulant n by n sign matrices and show that their spectra converge to the semicircular law when n grows. The Kolmogorov complexity of the proposed matrices equals to that of Golomb sequences and is at most 2log(n) bits.